IMPORTANT NOTE The following is an open-book, open-note final exam. Collaboration is not allowed. Please show all your work, as no credit will be given for unsupported answers. Feel free to e-mail me with questions at nega_ginge@yahoo.com. You can also call me at (225) 292-0475. You can either e-mail your solutions to me at the e-mail address above or send them by snail mail to Stephen Looney 11888 Longridge Ave. Apt. 1007 Baton Rouge, LA 70816 If you do use snail mail, please make sure that your solution to the final exam is post-marked by December 12, 2005. Good luck! Final Exam (Due Monday, December 12, 2005) NOTE: When giving the results for an hypothesis test, be sure to provide the value of the test statistic, the d.f. (if any), the p-value, and a conclusion. Be sure to indicate on the SAS output where you obtained your answers. Otherwise, I may not be able to give you any partial credit in case you make a mistake. In Exercises 1 & 2 below, use the exact procedure whenever one is available in SAS.

1. Consider Exercises 2.25 and 2.26, p. 50.

(i) Write the SAS code to test H0: OR = 1 vs. H0: OR > 1 and find a 95% CI(OR). (ii) Use the SAS output on pp. 2-3 below to perform the analyses specified in Part (i) above.

2

Exercise 2.25 The FREQ Procedure Statistics for Table of surgery by controlled Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1 0.5992 0.4389 Likelihood Ratio Chi-Square 1 0.5948 0.4406 Continuity Adj. Chi-Square 1 0.0860 0.7694 Mantel-Haenszel Chi-Square 1 0.5845 0.4445 Phi Coefficient 0.1209 Contingency Coefficient 0.1200 Cramer's V 0.1209 WARNING: 50% of the cells have expected counts less than 5. (Asymptotic) Chi-Square may not be a valid test. Pearson Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.5992 DF 1 Asymptotic Pr > ChiSq 0.4389 Exact Pr >= ChiSq 0.6384 Likelihood Ratio Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.5948 DF 1 Asymptotic Pr > ChiSq 0.4406 Exact Pr >= ChiSq 0.6384

(SAS Output continued on next page.)

3

Mantel-Haenszel Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.5845 DF 1 Asymptotic Pr > ChiSq 0.4445 Exact Pr >= ChiSq 0.6384 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 21 Left-sided Pr <= F 0.8947 Right-sided Pr >= F 0.3808 Table Probability (P) 0.2755 Two-sided Pr <= P 0.6384 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 2.1000 0.3116 14.1523 Cohort (Col1 Risk) 1.0957 0.8601 1.3957 Cohort (Col2 Risk) 0.5217 0.0973 2.7981 Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 2.1000 Asymptotic Conf Limits 95% Lower Conf Limit 0.3116 95% Upper Conf Limit 14.1523 Exact Conf Limits 95% Lower Conf Limit 0.2089 95% Upper Conf Limit 27.5522

Sample Size = 41

4

(iii) Assume that the data in Table 2.16 were obtained from a retrospective chart review. The investigator wants to submit a grant proposal so that she can conduct a much larger study. How many subjects would be required to achieve 80% power for detecting a difference in population proportions equal to the observed difference in sample proportions in these data? Assume that a significance level of α = .05 would be used in the eventual statistical analysis of the data from the larger study.

2. Consider Exercise 2.21, p. 49.

(i) Write the SAS code to test for a significant positive association between the row and column variables. (ii) Use the SAS output below and on pp. 5-6 to perform the test requested in Part (i) above.

Exercise 2.21 The FREQ Procedure Statistics for Table of mamm_exp by brca_det Statistic DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 4 24.1481 <.0001 Likelihood Ratio Chi-Square 4 26.7969 <.0001 Mantel-Haenszel Chi-Square 1 21.2957 <.0001 Phi Coefficient 0.2421 Contingency Coefficient 0.2353 Cramer's V 0.1712 WARNING: 22% of the cells have expected counts less than 5. (Asymptotic) Chi-Square may not be a valid test. Pearson Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 24.1481 DF 4 Asymptotic Pr > ChiSq <.0001 Exact Pr >= ChiSq 1.090E-04

(SAS Output continued on next page.)

5

Likelihood Ratio Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 26.7969 DF 4 Asymptotic Pr > ChiSq <.0001 Exact Pr >= ChiSq 2.519E-05 Mantel-Haenszel Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 21.2957 DF 1 Asymptotic Pr > ChiSq <.0001 Exact Pr >= ChiSq 2.989E-06 Statistic Value ASE ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Gamma 0.4460 0.0818 Kendall's Tau-b 0.2193 0.0402 Stuart's Tau-c 0.1666 0.0315 Somers' D C|R 0.1910 0.0359 Somers' D R|C 0.2518 0.0464 Pearson Correlation 0.2276 0.0401 Spearman Correlation 0.2344 0.0429 Lambda Asymmetric C|R 0.0000 0.0000 Lambda Asymmetric R|C 0.0000 0.0000 Lambda Symmetric 0.0000 0.0000 Uncertainty Coefficient C|R 0.0443 0.0156 Uncertainty Coefficient R|C 0.0333 0.0118 Uncertainty Coefficient Symmetric 0.0380 0.0134

(SAS Output continued on next page.)

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Pearson Correlation Coefficient ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Correlation (r) 0.2276 ASE 0.0401 95% Lower Conf Limit 0.1490 95% Upper Conf Limit 0.3062 Test of H0: Correlation = 0 ASE under H0 0.0428 Z 5.3185 One-sided Pr > Z <.0001 Two-sided Pr > |Z| <.0001 Exact Test One-sided Pr >= r 7.445E-07 Two-sided Pr >= |r| 2.989E-06 Sample Size = 412

3. Consider Exercises 3.10, p. 68, and Exercise 5.19, p. 139.

(i) Write the SAS code to carry out the analyses requested in Parts (a) – (d) of Exercise 3.10. (ii) Write the SAS code to fit a logistic regression model to these data, with “Group” and “Center” as explanatory

variables. Include the necessary options & statements to carry out the analyses requested in Parts (iv) (p. 13) & (vi) (p. 15) below. (iii) Use the SAS output on pp. 7-12 below to answer Parts (a) – (d) of Exercise 3.10. In Part (a), provide a 95% CI for the

conditional OR in each partial table and test the hypothesis that each conditional OR is equal to 1. (SAS Output begins on following page.)

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Exercise 3.10 CMH Analysis The FREQ Procedure Statistics for Table 1 of group by response Controlling for center=1 Pearson Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 3.3333 DF 1 Asymptotic Pr > ChiSq 0.0679 Exact Pr >= ChiSq 0.1698 Likelihood Ratio Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 3.4522 DF 1 Asymptotic Pr > ChiSq 0.0632 Exact Pr >= ChiSq 0.1698 Mantel-Haenszel Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 3.1667 DF 1 Asymptotic Pr > ChiSq 0.0752 Exact Pr >= ChiSq 0.1698 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 6 Left-sided Pr <= F 0.9901 Right-sided Pr >= F 0.0849 Table Probability (P) 0.0750 Two-sided Pr <= P 0.1698

8

Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 6.0000 0.8117 44.3512 Cohort (Col1 Risk) 3.0000 0.7864 11.4447 Cohort (Col2 Risk) 0.5000 0.2202 1.1351 Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 6.0000 Asymptotic Conf Limits 95% Lower Conf Limit 0.8117 95% Upper Conf Limit 44.3512 Exact Conf Limits 95% Lower Conf Limit 0.6027 95% Upper Conf Limit 79.9491 Sample Size = 20 Exercise 3.10 CMH Analysis The FREQ Procedure Table 2 of group by response Controlling for center=2 Pearson Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 2.2363 DF 1 Asymptotic Pr > ChiSq 0.1348 Exact Pr >= ChiSq 0.2657

9

Likelihood Ratio Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 2.3558 DF 1 Asymptotic Pr > ChiSq 0.1248 Exact Pr >= ChiSq 0.2657 Mantel-Haenszel Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 2.0643 DF 1 Asymptotic Pr > ChiSq 0.1508 Exact Pr >= ChiSq 0.2657 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 4 Left-sided Pr <= F 0.9837 Right-sided Pr >= F 0.1795 Table Probability (P) 0.1632 Two-sided Pr <= P 0.2657 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 6.6667 0.4866 91.3306 Cohort (Col1 Risk) 3.4286 0.5124 22.9400 Cohort (Col2 Risk) 0.5143 0.2035 1.2999

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Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 6.6667 Asymptotic Conf Limits 95% Lower Conf Limit 0.4866 95% Upper Conf Limit 91.3306 Exact Conf Limits 95% Lower Conf Limit 0.3370 95% Upper Conf Limit 392.4892 Sample Size = 13 Exercise 3.10 CMH Analysis The FREQ Procedure Table 3 of group by response Controlling for center=3 Pearson Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1.4462 DF 1 Asymptotic Pr > ChiSq 0.2291 Exact Pr >= ChiSq 0.3469 Likelihood Ratio Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1.4659 DF 1 Asymptotic Pr > ChiSq 0.2260 Exact Pr >= ChiSq 0.3469

11

Mantel-Haenszel Chi-Square Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 1.3611 DF 1 Asymptotic Pr > ChiSq 0.2433 Exact Pr >= ChiSq 0.3469 Fisher's Exact Test ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Cell (1,1) Frequency (F) 5 Left-sided Pr <= F 0.9555 Right-sided Pr >= F 0.2380 Table Probability (P) 0.1935 Two-sided Pr <= P 0.3469 Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control (Odds Ratio) 3.3333 0.4546 24.4428 Cohort (Col1 Risk) 1.8750 0.6441 5.4583 Cohort (Col2 Risk) 0.5625 0.2055 1.5395 Odds Ratio (Case-Control Study) ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Odds Ratio 3.3333 Asymptotic Conf Limits 95% Lower Conf Limit 0.4546 95% Upper Conf Limit 24.4428 Exact Conf Limits 95% Lower Conf Limit 0.3213 95% Upper Conf Limit 37.5956 Sample Size = 17

12

Exercise 3.10 00:30 Sunday, December 4, 2005 114 CMH Analysis The FREQ Procedure Summary Statistics for group by response Controlling for center Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 Nonzero Correlation 1 6.4242 0.0113 2 Row Mean Scores Differ 1 6.4242 0.0113 3 General Association 1 6.4242 0.0113 Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control Mantel-Haenszel 4.9158 1.4309 16.8873 (Odds Ratio) Logit 4.8887 1.4112 16.9348 Cohort Mantel-Haenszel 2.5399 1.1716 5.5064 (Col1 Risk) Logit 2.4098 1.1218 5.1767 Cohort Mantel-Haenszel 0.5226 0.3086 0.8849 (Col2 Risk) Logit 0.5209 0.3083 0.8800 Breslow-Day-Tarone Test for Homogeneity of the Odds Ratios ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.2369 DF 2 Pr > ChiSq 0.8883

Total Sample Size = 50

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(iv) Use the SAS output below and on pp. 14-15 answer the following questions. (a) Give the equation for the fitted logistic regression model. Clearly state what each variable in the model represents.

(b) Test the significance of each coefficient in the logistic regression model. Comment. (c) Find a 95% CI for the adjusted odds ratio for “Group” controlling for “Center” using the likelihood ratio method.

Exercise 3.10 Logistic Regression Analysis Model with Group & Center The LOGISTIC Procedure Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq group 1 6.4321 0.0112 center 2 0.4870 0.7839 Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -1.1884 0.4789 6.1588 0.0131 group 1 1.5965 0.6295 6.4321 0.0112 center 1 1 -0.0820 0.4235 0.0375 0.8464 center 2 1 -0.2227 0.4747 0.2201 0.6390

14

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits group 4.936 1.437 16.951 center 1 vs 3 0.679 0.165 2.795 center 2 vs 3 0.590 0.120 2.896 Association of Predicted Probabilities and Observed Responses Percent Concordant 63.5 Somers' D 0.419 Percent Discordant 21.7 Gamma 0.491 Percent Tied 14.8 Tau-a 0.208 Pairs 609 c 0.709 Profile Likelihood Confidence Interval for Parameters Parameter Estimate 95% Confidence Limits Intercept -1.1884 -2.2217 -0.3098 group 1.5965 0.4067 2.9005 center 1 -0.0820 -0.9290 0.7509 center 2 -0.2227 -1.1948 0.6961 Wald Confidence Interval for Parameters Parameter Estimate 95% Confidence Limits Intercept -1.1884 -2.1269 -0.2498 group 1.5965 0.3627 2.8303 center 1 -0.0820 -0.9120 0.7480 center 2 -0.2227 -1.1531 0.7077

15

Profile Likelihood Confidence Interval for Adjusted Odds Ratios Effect Unit Estimate 95% Confidence Limits group 1.0000 4.936 1.502 18.183 center 1 vs 3 1.0000 0.679 0.159 2.791 center 2 vs 3 1.0000 0.590 0.113 2.850 Wald Confidence Interval for Adjusted Odds Ratios Effect Unit Estimate 95% Confidence Limits group 1.0000 4.936 1.437 16.951 center 1 vs 3 1.0000 0.679 0.165 2.795 center 2 vs 3 1.0000 0.590 0.120 2.896

(v) The marginal analysis for this problem yielded OR = 4.75, with an exact 95% CI(OR) = (1.22, 19.47), and a p-value for the exact χ2 test of independence = 0.021. The logistic regression model containing Group as the only predictor yielded OR = 4.75, with a 95% CI(OR) using the likelihood-ratio method of (1.46, 17.10), and a p-value for the likelihood-ratio test of significance = 0.009. Does it appear that Center is a significant confounder of the association between Group and Response? Give reasons for your answer.

(vi) Use the SAS output on pp. 16-18 below to perform the likelihood ratio goodness-of-fit test and to assess model fit

using the diagnostic measures we discussed in Chapter 5. Does the logistic regression model containing Group and Center appear to provide a good fit to the data? Why or why not? (Note: The “Global Null Hypothesis” referred to in the SAS output is the hypothesis that all parameters in the logistic regression model except for α are equal to zero.) (SAS Output begins on following page.)

16

Exercise 3.10

Logistic Regression Analysis Model with Group & Center The LOGISTIC Procedure Deviance and Pearson Goodness-of-Fit Statistics Criterion Value DF Value/DF Pr > ChiSq Deviance 0.2356 2 0.1178 0.8889 Pearson 0.2371 2 0.1185 0.8882 Number of events/trials observations: 6 Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 70.029 68.714 SC 71.941 76.362 -2 Log L 68.029 60.714 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 7.3156 3 0.0625 Score 7.0746 3 0.0696 Wald 6.5764 3 0.0867

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Regression Diagnostics Pearson Residual Deviance Residual Covariates Case (1 unit = 0.03) (1 unit = 0.03) Number group center1 center2 Value -8 -4 0 2 4 6 8 Value -8 -4 0 2 4 6 8 1 1.0000 1.0000 0 0.1230 | | * | 0.1232 | | * | 2 0 1.0000 0 -0.1467 | * | | -0.1483 | * | | 3 1.0000 0 1.0000 0.1339 | | * | 0.1341 | | * | 4 0 0 1.0000 -0.1814 | * | | -0.1851 | * | | 5 1.0000 -1.0000 -1.0000 -0.2771 |* | | -0.2741 |* | | 6 0 -1.0000 -1.0000 0.2699 | | *| 0.2664 | | *| Regression Diagnostics Hat Matrix Diagonal Intercept group Case (1 unit = 0.05) DfBeta (1 unit = 0.06) DfBeta (1 unit = 0.05) Number Value 0 2 4 6 8 12 16 Value -8 -4 0 2 4 6 8 Value -8 -4 0 2 4 6 8 1 0.7516 | *| -0.0217 | * | 0.2007 | | * | 2 0.6466 | * | -0.2425 | * | | 0.2007 | | * | 3 0.7328 | *| 0.0608 | |* | 0.1466 | | * | 4 0.5098 | * | -0.2040 | * | | 0.1466 | | * | 5 0.6711 | * | 0.0580 | |* | -0.3617 |* | | 6 0.6881 | * | 0.4993 | | *| -0.3617 |* | | Confidence Interval Displacement C center1 center2 Case DfBeta (1 unit = 0.04) DfBeta (1 unit = 0.04) (1 unit = 0.03) Number Value -8 -4 0 2 4 6 8 Value -8 -4 0 2 4 6 8 Value 0 2 4 6 8 12 16 1 0.2893 | | *| -0.1504 | * | | 0.1842 | * | 2 -0.2102 | * | | 0.0724 | | * | 0.1114 | * | 3 -0.1968 | * | | 0.3315 | | *| 0.1840 | * | 4 0.1027 | | * | -0.2028 | * | | 0.0698 | * | 5 0.2502 | | * | 0.2519 | | * | 0.4764 | * | 6 -0.2489 | * | | -0.1932 | * | | 0.5150 | *|

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Regression Diagnostics Confidence Interval Displacement CBar Delta Deviance Delta Chi-Square Case (1 unit = 0.01) (1 unit = 0.01) (1 unit = 0.01) Number Value 0 2 4 6 8 12 16 Value 0 2 4 6 8 12 16 Value 0 2 4 6 8 12 16 1 0.0458 | * | 0.0609 | * | 0.0609 | * | 2 0.0394 | * | 0.0613 | * | 0.0609 | * | 3 0.0492 | * | 0.0672 | * | 0.0671 | * | 4 0.0342 | * | 0.0685 | * | 0.0671 | * | 5 0.1567 | *| 0.2318 | *| 0.2335 | *| 6 0.1606 | *| 0.2316 | *| 0.2335 | *|

4. Use the results in the following table to answer Exercise 5.31, p. 142. (This table is similar to Table 5.8, p. 128.) Calculate the

entries in the “Difference” and “Difference df” columns and then use SimCalc, StatCalc, Excel or other software to calculate the p-value at each stage of the elimination procedure. In the “Action” column, state the action you took to get from the previous model to the current one. I have done the 1st model comparison for you as an example.

Model

Predictors

Deviance

df

Models

Compared

Difference

Difference

df

P-value

Action

(1)

C*S*W

167.93

152

--

--

--

--

(2)

C*S + C*W + S*W

167.94

155

(2) vs. (1)

.01

3

1.000

Drop C*S*W

(3a)

C*S + S*W

174.79

158

(3a) vs. (2)

(3b)

C*W + S*W

176.97

161

(3b) vs. (2)

(3c)

C*S + C*W

169.77

157

(3c) vs. (2)

(Table continued on next page.)

19

Model

Predictors

Deviance

Df

Models

Compared

Difference

Difference

Df

P-value

Action

(4a)

S + C*W

179.28

163

(4a) vs. (3c)

(4b)

W + C*S

176.50

160

(4b) vs. (3c)

(5)

C + S + W

187.00

166

(5) vs. (4b)

(6a)

C + S

208.83

167

(6a) vs. (5)

(6b)

S + W

195.33

169

(6b) vs. (5)

(6c)

C + W

188.54

168

(6c) vs. (5)

(7a)

C

212.06

169

(7a) vs. (6c)

(7b)

W

195.74

171

(7b) vs. (6c)

(8)

D + W

189.17

170

(8) vs. (7b)

(9)

None

225.76

172

(9) vs. (8)